This paper deals with the quantitative Schmidt’s subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt’s subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions.
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Cartan, H.: Sur les zéroes des combinaisons linéaries de p fonctions holomorphes données. Mathematica 7, 80–103 (1933)
Chen, Z., Ru, M., Yan, Q.: The truncated second main theorem and uniqueness theorems. Sci. China Math. 53, 605–616 (2010)
Chen, Z., Ru, M., Yan, Q.: The degenerated second main theorem and Schmidt’s subspace theorem. Sci. China Math. 55, 1367–1380 (2012)
Corvaja, P., Zannier, U.: On a general Thue’s equation. Am. J. Math. 126, 1033–1055 (2004)
Evertse, J.H., Ferretti, R.G.: Diophantine inequalities on projective varieties. Int. Math. Res. Not. 25, 1295–1330 (2002)
Evertse, J.H., Ferretti, R.G.: A generalization of the subspace theorem with polynomials of higher degree. In: Tichy, R.F., Schlickewei, H.P., Schmidt, K. (eds.) Diophantine Approximation, Festschrift for Wolfgang Schmidt, pp. 175–198. Springer, Vienna (2008)
Evertse, J.H., Schlickewei, H.P.: A quantitative version of the absolute subspace theorem. J. Reine Angew. Math. 548, 21–127 (2002)
Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry, vol. II. Cambridge University Press, Cambridge (1952)
Giang, L.: On the quantitative subspace theorem. J. Number Theory 145, 474–495 (2014)
Giang, L.: An explicit estimate on multiplicity truncation in the degenerated second main theorem. Houst. J. Math. 42, 447–462 (2016)
Lei, S., Ru, M.: An improvement of Chen-Ru-Yan’s degenerated second main theorem. Sci. China Math. 58, 2517–2530 (2015)
Nochka, E.I.: On the theory of meromorphic functions. Sov. Math. Dokl. 27, 377–381 (1983)
Noguchi, J.: A note on entire pseudo-holomorphic curves and the proof of Cartan–Nochka’s theorem. Kodai Math. J. 28, 336–346 (2005)
Osgood, C.F.: A number theoretic-differential equations approach to generalizing Nevanlinna theory. Indian J. Math. 23, 1–15 (1981)
Osgood, C.F.: Sometimes effective Thue–Siegel–Roth–Schmidt–Nevanlinna bounds, or better. J. Number Theory 21, 347–389 (1985)
Quang, S.D.: Schmidt’s subspace theorem for moving hypersurfaces in subgeneral position. Int. J. Number Theory 14(1), 103–121 (2018)
Quang, S.D.: A generalization of the subspace theorem for higher degree polynomials in subgeneral position. Int. J. Number Theory 15(4), 775–788 (2019)
Quang, S.D.: Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces. Trans. Am. Math. Soc. 371(4), 2431–2453 (2019)
Ru, M., Stoll, W.: The second main theorem for moving targets. J. Geom. Anal. 1, 99–138 (1991)
Ru, M., Stoll, W.: The Cartan conjecture for moving targets. In: Proceedings of Symposia in Pure Mathematics, vol. 52, Part 2, pp. 477–508. American Mathematical Society (1991)
Ru, M.: On a general form of the second main theorem. Trans. Am. Math. Soc. 349, 5093–5105 (1997)
Ru, M.: A defect relation for holomorphic curves intersecting hypersurfaces. Am. J. Math. 126, 215–226 (2004)
Ru, M.: Holomorphic curves into algebraic varieties. Ann. Math. 169, 255–267 (2009)
Schmidt, W.M.: Norm form equations. Ann. Math. 96, 526–551 (1972)
Schmidt, W.M.: Simultaneous approximation to algebraic numbers by elements of a number field. Monatsh. Math. 79, 55–66 (1975)
Schmidt, W.M.: Diophantine Approximation. Lecture Notes in Mathematics, vol. 785. Springer, Berlin (1980)
Schmidt, W.M.: The subspace theorem in Diophantine approximation. Compos. Math. 96, 121–173 (1989)
Shirosaki, M.: Another proof of the defect relation for moving targets. Tôhoku Math. J. 43, 355–360 (1991)
Vojta, P.: Diophantine Approximation and Value Distribution Theory. Lecture Notes in Mathematics, vol. 1239. Springer, Berlin (1987)
Vojta, P.: A refinement of Schmidt’s subspace theorem. Am. J. Math. 111, 489–518 (1989)
Vojta, P.: On Cartan’s theorem and Cartan’s conjecture. Am. J. Math. 119, 1–17 (1997)
We would like to thank the referee for carefully reading our manuscript and for his/her valuable comments on the first version of this paper, which help us to improve the quality of the paper.
Dedicated to Professor Le Mau Hai on the occasion of His 70th birthday.
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Si, D.Q. Quantitative subspace theorem and general form of second main theorem for higher degree polynomials. manuscripta math. (2021). https://doi.org/10.1007/s00229-021-01329-z
Mathematics Subject Classification
- Primary 11J68
- Secondary 11J25